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Negative Hyperfine
Thoughts about why hyperfine is positive and why it is independent of the gyromagnetic ratios of the interacting spins. http://arxiv.org/pdf/1408.1347.pdf (Si29 paper) This paper has two nuclei used, one with a +ve and one with a -ve gyromagnetic ratio (GMR). Given Feynman's logic that A is a product of magnetic moments of electron and nucleus, then the sign should change when the sign of the GMR changes. However, both nuclei are shown to have positive hyperfines in this paper and no minus sign is introduced in the formulae to rectify any issue. A reference is given to (http://journals.aps.org/pr/pdf/10.1103/PhysRev.114.1219) to justify the formula used which presumes a positive GMR for electron and phosphorus, but then uses a sign convention in the Hamiltonian to rectify the difference. http://journals.aps.org/prl/pdf/10.1103/PhysRevLett.113.246801 (final copy) In the published version the equation is rearranged to have all positive terms in the Hamiltonian and the electron GMR (ɣ) is redefined as negative. Notably, the published version also adds absolute value over the A_Si term when quoting it as 2.205(5) MHz. http://beyondbinary.wikia.com/wiki/File:Pla_Si29_Hamiltonian.pngScreenshot from Pla2014 PRL paper, showing positive hyperfine terms for both nuclei One intuitive explanation for why the hyperfine doesn't 'go negative' could be that hyperfine is best understood from the perspective of the electron, in which case the GMR of the nucleus is not important, only the field caused by its spin. However, the GMR is what tells us the field caused by its spin and a -ve GMR implies that a flip occurs and hence a negative hyperfine term (A) should be used to account for this flip. Steger2011 follows a convention that the Hamiltonian uses different signs for electron and nucleus and allows them both to have positive GMR's, while the hyperfine is left positive. As below: This can be easily translated into the consistent, conventional "E=-μB" form by allowing the electron GMR to be defined by its true negative value, giving: Which removes any intrinsic need for a negative hyperfine here. It seems clear from these two sources that negative hyperfines are not necessary even when the nuclei involved have opposite signs in their GMR. It seems a bit odd to me that an electron should experience a shift in energy in the same direction due to two nuclei in the same spin state but with opposite signed GMR's, because this implies that the magnetic moment of each is in opposite directions and hence one would expect that the shift felt by the electron would be different. The only explanation I can think of now is that hyperfine interaction always favours spins that anti-align, regardless of the GMR's. In this sense the hyperfine interaction would be best considered as a mutual interaction and not looked at from either the perspective of the electron or the nucleus, but the joint system desiring to minimise it's net spin. This goes against Feynman's logic in describing the hyperfine as a product of magnetic moments, but even then in his hydrogen example the logic failed because it was a case of opposite signs and the hyperfine of hydrogen is experimentally found to be positive. Feynman notes[1] "(We originally took A as positive because the theory we spoke of says it should be, and experimentally it is indeed so.)" despite earlier noting "μ_p is the magnetic moment of the proton, which is about 1000 times smaller than μ_e, and has the opposite sign". Interestingly, the lecture goes on to show how the hyperfine ground state of hydrogen (E=-4A) has no magnetic moment and would pass through a Stern-Gerlach apparatus undeflected. The state is calculated as the singlet state: Singlet states are known to have zero magnetic moment, but when you consider the opposite signs of the GMR's then the reason behind this is no longer so obvious. Shouldn't anti-aligned particles with opposite GMR's actually give a net magnetic moment? Perhaps the issue is that if a singlet state did have a magnetic moment, it would be completely undefined in direction since the singlet is a joint state with complete symmetry, hence the magnetic moment cannot be asymmetric either and this joint magnetic moment is cancelled by a superposition of states with the joint-moment in opposite directions. Hence why the T0 triplet state still has a net magnetic moment, despite also being comprised of a superposition of anti-aligned states. It seems that negative hyperfine constants are not necessary, since the main motivation was to give an intuitive picture of hyperfine coupling in terms of magnetic moments. But it seems that this intuitive picture is out the window. But what would happen if hydrogen had a negative GMR? (like 3He) A nucleus with a negative GMR and an electron with a negative GMR, both producing a magnetic moment opposite to their spin, so when their positive hyperfine term dominates they form a ground state with opposite spins (singlet) and their magnetic moments are naturally opposed also. But what is the relevant magnetic field for each moment to interact with? The value of the GMR doesn't set an intrinsic field, so neither particle can evaluate the effective frequency shift caused by the other intrinsically. All that can be discussed is their coupling and how their energy levels split when they couple in the absence of an external field: the hyperfine coupling. In this case their moments anti-align and this would cause a paradox, since each spin wants to be anti-parallel to the moment of the other spin (equivalent of spin-'down' ground state) and each spin is defined as opposite to its own moment. Hence, both spins want to 'ferromagnetically' combine their moments, but this implies that they align their spins, which defies the definition of the singlet state and hyperfine coupling with positive hyperfine. A negative hyperfine would solve this problem when the two particles have GMR's of the same value (argument for both positive is similar except spins and moments align and particles desire to align with external fields - goes against hyperfine being positive). So it seems that a negative hyperfine is only needed when the GMR of the nucleus is negative. Which was the case for Si29 in the paper above, but the hyperfine was seen as positive... Need to come back to this in the future. Notes: The singlet state's missing magnetic moment is very interesting, perhaps this could tie in to magnetic monopole theories in the anti-verse, whereby the information of the individual states of the singlet is lost to our universe, and essentially has entered a black-hole in the heart of the Torus (see Toroidal Photon Model) that on the 'other side' can form a complementary anti-particle with the same macroscopic properties. This isn't a magnetic monopole as I've described it... perhaps that was a bit over-zealous, but an anti-hydrogen in the anti-verse would be the simplest resolution. Although, this then gives no solution to the missing magnetic moment (which was the logic behind suggesting a magnetic monopole). References #↑ Feynman Lecture on Hydrogen Hyperfine http://www.feynmanlectures.caltech.edu/III_12.html Category:Theories Category:Quantum Category:Physics